Twistor theory of the Chen--Teo gravitational instanton

Toric Ricci–flat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, for the five–parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen–Teo family contains a two–parameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form.